1. **Problem Statement:** Verify that $(x+3)$ is a factor of $f(x) = 3x^3 + 2x^2 - 19x + 6$, find the remaining factors, write the complete factorization, and list all real zeros. 2. **Step 1: Verify the factor $(x+3)$** Use the Factor Theorem: if $(x+3)$ is a factor, then $f(-3) = 0$. Calculate: $$f(-3) = 3(-3)^3 + 2(-3)^2 - 19(-3) + 6 = 3(-27) + 2(9) + 57 + 6 = -81 + 18 + 57 + 6 = 0$$ Since $f(-3) = 0$, $(x+3)$ is indeed a factor. 3. **Step 2: Find the remaining factor(s)** Divide $f(x)$ by $(x+3)$ using polynomial long division or synthetic division. Using synthetic division: - Write coefficients: $3, 2, -19, 6$ - Use root $-3$: $$\begin{array}{r|rrrr} -3 & 3 & 2 & -19 & 6 \\ & & -9 & 21 & -6 \\ \hline & 3 & -7 & 2 & \cancel{0} \\ \end{array}$$ The quotient is $3x^2 - 7x + 2$. 4. **Step 3: Factor the quadratic $3x^2 - 7x + 2$** Find two numbers that multiply to $3 \times 2 = 6$ and add to $-7$: these are $-6$ and $-1$. Rewrite: $$3x^2 - 6x - x + 2 = 3x(x - 2) -1(x - 2) = (3x - 1)(x - 2)$$ 5. **Step 4: Write the complete factorization** $$f(x) = (x + 3)(3x - 1)(x - 2)$$ 6. **Step 5: List all real zeros** Set each factor equal to zero: - $x + 3 = 0 \Rightarrow x = -3$ - $3x - 1 = 0 \Rightarrow x = \frac{1}{3}$ - $x - 2 = 0 \Rightarrow x = 2$ So the real zeros are: $$x = -3, \frac{1}{3}, 2$$ 7. **Step 6: Confirm with graphing utility** The graph crosses the x-axis at these points, confirming the zeros. Final answers: - Remaining factors: $(3x - 1), (x - 2)$ - Complete factorization: $f(x) = (x + 3)(3x - 1)(x - 2)$ - Real zeros: $-3, \frac{1}{3}, 2$